On the $\Lambda$-fractional continuum mechanics fields

Abstract

After defining the fractional $\varLambda$-derivative, having all the prerequisites for corresponding to a differential, the fractional $\varLambda$-strain has already been established. Furthermore, only globally variational principles are allowed in the context of fractional analysis. Hence, balance laws, yielding the various field equations in $\Lambda$-fractional continuum mechanics, are derived, allowing corners in their fields. The basic balance laws of mass, linear and rotational momentum, and energy conservation with jump conditions are derived in the context of $\Lambda$-fractional analysis.

Appendix

1.1 $\Lambda$-Fractional transformation

The present appendix is a concise chapter dealing with the $\Lambda$-fractional analysis. As it has already been pointed out, the (x, y(x)) initial space is transformed into the (X, Y(X)) $\Lambda$-fractional space through the transformation,

$$\begin{aligned}{} & {} X={ }_{\alpha }^{RL} \textrm{I}_{x}^{1-\gamma } x \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} Y\left( X \right) =\textrm{I}^{1-\gamma }\left( {y\left( {x\left( X \right) } \right) } \right) . \end{aligned}$$

(A.2)

Repeating that fractional derivatives fail to generate differential geometries, their use is not appropriate for describing non-local theories in various fields.

Let us consider the well-known R–L fractional derivative, of a function f(x) for 0< $\gamma<$ 1,

$$\begin{aligned} {}_{a}^{RL} D_{x}^{\gamma } y(x)=\frac{d}{dx}({ }_{\text{ a }}I_{x}^{1-\gamma } \;y(x)) \end{aligned}$$

(A.3)

Considering, (A.1), (A.2) the initial space (x, y(x)) is transformed into the $\Lambda$-fractional space (X, Y(X)).

Although, the R–L derivative might be satisfactory, for the present purposes,, variational problems might be raised with its use. Indeed, Noether’s variational theory, Atanackovic et al. [23]., might be questionable since the fractional order differentials are mixed with fractional order differentials and the adoption of that derivative might not satisfy geometrical and physical demands, The $\Lambda$-Fractional Derivative introduced by Lazopoulos [18], could be adopted without the problems that the R–L exhibits. The $\Lambda$-Fractional Derivative has been defined by the ratio of the y(x) over the R–L fractional derivative of x:

$$\begin{aligned} { }_{a}^{\Lambda } D_{x}^{\gamma } y\left( x \right) =\frac{{ }_{a}^{RL} D_{x}^{\gamma } y\left( x \right) }{{ }_{a}^{RL} D_{x}^{\gamma } x}=\frac{d\left( {{ }_{a}^{RL} \textrm{I}_{x}^{1-\gamma } \left( {y\left( x \right) } \right) } \right) }{d\left( {{ }_{a}^{RL} \textrm{I}_{x}^{1-\gamma } x} \right) } \end{aligned}$$

(A.4)

Indeed, the $\Lambda$-fractional derivative, Eq. (A.4) may be expressed as a common derivative of the function YX) concerning X. Hence the analysis has been transferred to the $\Lambda$-space (X, Y(X)) from the initial space (x, y(x)). That transformation exhibits two major advantages. First, the non-local problem in the initial space becomes local in the $\Lambda$-fractional space, and second the $\Lambda$-fractional derivative access a local character conforming with the conventional local differential analysis. Hence differential geometry is developed in the $\Lambda$-fractional space. Applications of geometry in various fields of physics, mechanics, etc., may be performed based upon that differential geometry. Then, the results may be transferred back to the initial variables. Indeed, the transferring may be performed through the relationship, through the inverse $\Lambda$-fractional transformation,

$$\begin{aligned} {}_{\alpha } \textrm{I}_{x}^{\gamma -1}\left( {}_{\alpha }^{RL} D_{x}^{\gamma -1}\left( y\left( x \right) \right) \right) =y(x). \end{aligned}$$

(A.5)

Following that procedure, various fractional mathematical analysis areas, such as Fractional Differential Geometry, fractional field theory, fractional differential equations, etc may be established. The proposed $\Lambda$-fractional derivative satisfies Leibnitz’s rule for the product of the $I^{1-a}f(x)$ derivatives and $I^{1-a}y(x)$ the rule of the composite derivation. Indeed, Leibnitz’s rule concerning the derivative of the product of the functions yields,

$$\begin{aligned} \frac{d\left( {D^{\gamma -1}f(x)\cdot D^{\gamma -1}y(x)} \right) }{d\left( {D^{\gamma -1}x} \right) }=\frac{d\left( {D^{\gamma -1}f(x)} \right) }{d(D^{\gamma -1}x)}D^{\gamma -1}y(x)+D^{\gamma -1}f(x)\frac{d\left( {D^{\gamma -1}y(x)} \right) }{d\left( {D^{\gamma -1}x} \right) } \end{aligned}$$

(A.6)

Furthermore, the fractional derivative of a composite function is defined by,

$$\begin{aligned} \frac{d\left( {D^{\gamma -1}f(y(x))} \right) }{d(D^{\gamma -1}x)}=\frac{d\left( {D^{\gamma -1}f(y)} \right) }{d\left( {D^{\gamma -1}y} \right) }\cdot \frac{d\left( {D^{\gamma -1}y\left( x \right) } \right) }{d\left( {D^{\gamma -1}x} \right) } \end{aligned}$$

(A.7)

The linearity properties along with Leibnitz’s and composition rules, Eqs. (A.6) and (A.7), satisfy all the Differential Topology prerequisites, Chillingworth [17], for the $\Lambda$-fractional derivatives. Hence the $\Lambda$-fractional derivatives are well-established mathematical quantities acquiring the mathematical characteristics of the real derivatives. Hence, differential geometry may be established in the $\Lambda$-fractional space. The results may be transferred into the initial space through the inverse $\Lambda$-fractional transformation. The non-local mathematical analysis is established with derivatives having non-local character.